Gaunt term

Note that the subscript on the a matrices refers to the particle, and a here includes all of the tlx, tty and components in eq. (8.4). The first correction term in the square brackets is called the Gaunt interaction, and the whole term in the square brackets is the Breit interaction. The Dirac matiices appear since they represent the velocity operators in a relativistic description. The Gaunt term is a magnetic interaction (spin) while the other term represents a retardation effect. Eq. (8.27) is more often written in the form... 

Results from fully relativistic calculations are scarce, and there is no clear consensus on which effects are the most important. The Breit (Gaunt) term is believed to be small, and many relativistie calculations neglect this term, or include it as a perturbational term... 

For molecules the evaluation of the Breit correction to the Coulomb-type electron-electron interaction operator becomes computationally highly demanding and cannot be routinely evaluated, not even on the Dirac-Fock level. To test the significance of the Breit interaction, the Gaunt term is evaluated as a first-order perturbation. It turned out that it can be neglected in most cases as can be seen from the DF 4- Bmag calculations cited in Table 2.1. 

Most 4-component relativistic molecular calculations are based on the Dirac-Coulomb Hamiltonian corresponding to the choice g = Coulomb The Gaunt term of (173) has been written in a somewhat unusual manner. The speed of light has been inserted in the numerator which clearly displays that the Gaunt term has the form of a current-current interaction, contrary to the... 

Here, the current density j r) is still required, but it is now uniquely determined by the electron density, j = j[p]. If magnetic interactions between the electrons are not included — that is, when the Dirac-Coulomb Hamiltonian is employed and the Breit or Gaunt terms are omitted — then also the DFT analog of the Coulomb interaction will reduce to a functional J[p] of the density only [396]. Moreover, it now becomes possible to set up theories in which not the full 4-current is used as the fundamental variable, but only those parts of it that are related to the total electron density and the spin density. 

The integrals over this operator depend on densities over a or a x r. The integrals over the first (Gaunt) term are the same as the regular Coulombic electron repulsion integrals, while the second term requires calculation of integrals that are not encountered in nonrelativistic calculations. If we restrict our discussion to contributions from the Gaunt operator, we get densities of the form... 

The modified operators for the Gaunt and Breit interactions can be derived in an analogous manner. The derivation is somewhat more involved than for the Coulomb interaction due to the presence of the a matrices. Here we derive the Gaunt terms, but the gauge term for the Breit interaction is considerably more complicated (and the derivation may be found in appendix F). 

The gauge term, which comprises the difference between the Gaunt and the Breit interactions (see (5.48) and (5.49)), is more complicated than the Gaunt term due to the scalar quadruple product involving the alpha mafiices ... 

The two-electron operator is given in the nuclear frame and not in the reference of either electron. The spin-orbit coupling due to the relative motion of elecrons therefore splits into two parts The total interaction is the coupling of the spin of a selected reference electron with the magnetic field induced by a second electron. The spin-same orbit (SSO) and spin-other orbit (SOO) contributions arise from the motion of the reference electron and the other electron, respectively, relative to the nuclear frame and are carried by the Coulomb and Gaunt terms, respectively. For most molecular application it suffices to include the Coulomb term only, thus defining the Dirac-Coulomb Hamiltonian, but for the accurate calculation of molecular spectra the Gaunt term should be included as well. 

Reference values for the various 2-component relativistic Hamiltonians are provided by the 4-component Dirac-Coulomb Hamiltonian, but we have also included orbital energies obtained with the Dirac-Coulomb-Gaunt (DCG) Hamiltonian. As already mentioned, the Gaunt term brings in spin-other-orbit (SOO) interaction. Since spin-orbit interaction induced by other electrons will oppose the one induced by nuclei we see from Table 3.3 that the spin-orbit splitting of orbital levels is overall reduced. However, one should note that the Gaunt term also modifies /2 levels. 

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